The statistics of electron–hole avalanches

Charge multiplication through avalanche processes is commonly employed in the detection of single photons or charged particles in high-energy physics and beyond. In this report, we provide a detailed discussion of the properties of avalanches driven by two species of charge carriers, e.g. electrons and holes in a semiconductor exposed to an electric field. We derive equations that describe the general case of avalanches developing in position-dependent electric fields and give their analytical solutions for constant fields. We discuss consequences for the time resolution achievable with detectors that operate above the breakdown limit, e.g. single-photon avalanche diodes (SPADs) and silicon photomultipliers (SiPMs). Our results also describe avalanches that achieve finite gain and are important for avalanche photodiodes (APDs) and low-gain avalanche detectors (LGADs).